Wavelet solution of degenerate parabolic equations arising in stochastic volatility models

Ana-Maria Matache

Abstract: The analysis and implementation of deterministic pricing schemes for general contracts within stochastic volatility models is addressed. While for European Vanilla options closed form expressions for the prices are available, other types of contracts as, e.g., compound style options have to be priced numerically. Our approach is based on a Galerkin Finite Element (FE) discretization in the asset price variable and in the stochastic volatility variable of the associated deterministic partial differential equation (PDE). This parabolic PDE has, in general, degenerate coefficients.

Upon discretization, the linear systems to be solved in each implicit time step have ill-conditioned stiffness matrices.

We propose here a tensor product spline wavelet basis for the discretization in the asset price variable and in the stochastic volatility variable which allows, by the multiresolution analysis in weighted Sobolev spaces due to Beuchler, Schneider and Schwab (2002) for efficient preconditioning. Numerical examples for the Heston model and for models when the stochastic volatility is a function of a mean-reverting Orstein-Uhlenbeck process are presented.

References: [1] Beuchler, S., Schneider, R., Schwab, C. (2002): Multiresolution weighted norm equivalences and applications, Research Report No. 2002-13, Seminar für Angewandte Mathematik, ETH Zürich.